By two
(equal) arcs a lens or lemon
curve can be formed. This is a curvilinear
polygon.
This is an old symbol, denoting a fish as well as the female vulva.
For Pythagoras it was the symbol of the intersection of the world of the divine
(one circle) with the world of matter (other circle).
Later the sign was adopted by the Christians, with names as:

Extending the arcs in one side results in the Christian ichtus symbol.
An annulus is the form of two concentric circles, the shape of a ring.

A more recent use is a pattern formed in the shape of two circle arcs, the stamp of Truchet. The pattern
consists of two different orientations of the stamp next to each other, in an arbitrary sequence orientation.
These patterns have been used in the construction of games, e.g. the 'black path game' and 'meander'.
The pattern was found by Cyril Stanley Smith, who wrote an article 2)
about Truchet. Smith was a metallurgist and historian of technology at MIT who studied crystals.
The pattern described by Truchet was the Truchet tiling.

Three circlesA circular
triangle is formed by three (equal) arcs.
In the case of three equal arcs beginning in the angles of an equal
triangle, it is called the Reuleaux triangle.
This triangle can be rotated inside a square,
so making it possible to drill a hole with (nearly) the shape of a square.

Archimedes described the arbelos
or shoemaker's knife. The curve
resembles the blade of a knife used by ancient cobblers.
It appears that the arbelos has some remarkable qualities, for instance:

the arc length along the
enclosing semicircle is the same as along the two smaller semicircles

draw a perpendicular from the tangent of the two semicircles to the
edge of the large circle. Then the area of the circle with as diameter the
perpendicular is
the same as the area of the arbelos

Now, let's denote the left semicircle as C, with center (-c, 0) and the right
semicircle as D, with center (d,0). C and D touch at O, E is the large
semicircle.
Now define semicircles C(nc) and D(nd), with centers (-nc,0), (nd,0)
respectively, which touch in O. Then
Schoch's circle S is the circle that touches
E internally and C(2c) and D(2d) externally.
Woo considered the Schoch line w through the
center of the Schoch's circle and parallel to the y-axis.
Now it appears that the circles with center on the Schoch line and touching C(nc)
and D(nd) externally are also Archimedean circles,
and they are called Woo
circles.

Thomas Schoch and Peter Woo studied these circles, H. Okumura and M. Watanabe
made a generalization of the circles.

The Pappus chain is made by constructing
tangent Pappus circles in the arbelos, from right to left in the picture.
The centers of the circles lie on an ellipse.
The Bankoff circle B is the circle through
the cusp of the arbelos and the tangent points of the first
Pappus circle.Leon Bankoff (1908-1997) was an American dentist and mathematician. He
became a prominent Los Angeles dentist with famous patients of the Beverly
Hills-Hollywood scene.

Given a triangle, put
three circles in it, so that each circle is tangent to the other two and to two
sides of the triangle. These
circles are the Malfatti circles.
These were for many years considered to provide a solutions for the so-called
Malfatti´s problem, later it appeared they are not.Giovanni Francesco Malfatti (1731-1807) was an Italian
mathematician in Ferrara.

Four circlesArchimedes constructed the salinon as a
combination of four semicircles. The centers of the circles are lying on a line,
the two small circles are of equal size. Archimedes described the curve,
that is also called the salt-cellar4),
in his Book of Lemmas.

More concentric circles can be found at a target for shooting or darting.
In the 16th century scientists as Giordano Bruno and John Dee looked at
the spiritual meaning of so-called hermetic forms, which often contained
circles. By
placing those circles in the right place a resonance with his celestial (pure) counterpart
could be reached. An example of these forms is - with four circle arcs - John Dee's magic hieroglyphic5).

The circle
skating rink curve is formed by four circle arcs, one four each side, so
that the curve is tangent to two concentric circles (of different size).
The four arcs are from circles with the same size, but different center.

An application of the profile has been found in engine
design. The Quasiturbine (or
Qurbine) is a rotating engine with a
rotor,
trapped inside an internal housing contour, which does not require a central shaft
or support. This concept brings the engine dead time to zero.
The method of operation allows the engine to be extremely flexible. The
developer claims that it is possible to use it for pneumatics, hydraulics,
steam, petrol, natural gas and hydrogen.

The precise formula is not yet available to me, it can be found in the patent
for this engine.
Five circles
A combination of five
circles we can see in the symbol of the car Audi - circles placed on one line - and in the
symbol of the Olympic games - the five continents, the circles now positioned in two
lines. Is this similarity the explanation for the sponsorship of Audi of the Dutch Olympic
team?
A circle placed around a cross is the symbol of the Ku-klux klan 6)
movement.

Infinite circlesAn infinite sequence of tangent
circles for which the radius converges to zero, is called a Hawaiian
earring.
The radius can have forms like 1/n (see picture) or 1/n2 (n=1,2,
...).
The topology of this set and its generalizations has been studied by Eda
and Kawamura.